Sci & Educ (2010) 19:91–113 DOI 10.1007/s11191-008-9183-1

On the Concept of Force: How Understanding its History can Improve Physics Teaching Ricardo Lopes Coelho

Published online: 14 January 2009
Springer Science+Business Media B.V. 2009

Abstract Some physicists have pointed out that we do not know what force is. The most common definition of force in textbooks has been criticized for more than two centuries. Many studies have shown that the concept of force is a problem for teaching. How to conceive force on the basis of the concepts and criticism of force in the works of Newton, Euler, d’Alembert, Lagrange, Lazare Carnot, Saint-Venant, Reech, Kirchhoff, Mach, Hertz and Poincare´ is the question of the present article. This part of the article is followed by an overview of definitions of force in contemporary textbooks. In the next part, an answer to the question is given: how to understand force within the framework of the laws of motion and in applications. Finally, some educational implications are considered.

1 Introduction In the International Congress for Philosophy in Paris, 1900, Poincare´ put forward the question of if the fundamental equation of dynamics, F = ma, is verifiable experimentally. The question itself involves, however, a problem, he said, for we do not even know what force and mass are. In recent textbooks on mechanics, as for instance in Bergmann and Schaefer’s, Experimental Physics (1998) or Dransfeld et al. Physics (2001), it can also be read that we do not know what force is. If we do not know what it is, it is difficult to explain it in the best way. As force is a fundamental concept of mechanics and mechanics is basic in physics, it is not surprising that force is the dominant theme in the misconceptions’ literature (Carson and Rowlands 2005, p. 473). In textbooks of the twentieth and twenty-first century, force is in general defined as the cause of acceleration. Since acceleration is observable, its cause must be something real. Thus, force is real. Some physicists and philosophers of science have, however, pointed out that force does not exist in reality. Anyway, the mere fact that these two kinds of theses coexist, shows the difficulty in ‘‘seeing’’ force in phenomena. Thus, the abstraction of force R. L. Coelho (&) Faculty of Science, University of Lisbon, Campo Grande C4, 1749-016 Lisbon, Portugal e-mail: [email protected]

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from accelerated motions must obviously be difficult for students (see Carson and Rowlands 2005, p. 479; Rowlands et al. 2007, p. 30–31; Matthews 2008, p. 7, 10). There was a considerable effort concerning the understanding of force. D’Alembert, Lazare Carnot, Kirchhoff, Hertz, among others, did not only criticize the most common definition of force but also developed new theories in order to avoid that concept. Otherwise, there was a considerable effort as well in systematizing and applying mechanics to new domains—Newton, Euler, Lagrange, among many others—which is connected with that concept of force. How force could be conceived in compliance with these scientists’ contributions and without the inconveniences raised by the criticism of the concept, is the question to deal with in the present article. To this aim, the authors who have been object of historical studies on the concept of force (Dugas 1950; Jammer 1999; Coelho 2001) will be considered. This part is followed by an overview of definitions of force in contemporary textbooks (1901–2008). Finally, the connection between the concept, phenomena and equation of force will be dealt with.

2 History In 1687, Newton published the Mathematical Principles of Natural Philosophy. The fundaments of his theory consist of eight definitions and three axioms. The first definition concerns matter, the second motion and the other six concern forces. Five of the eight propositions define quantities and the other three, concepts: innate, impressed and centrifugal force (Definition III, IV, V). Centrifugal force is a particular case of impressed force.1 Thus, according to the definitions, there are two kinds of force: innate and impressed. Innate force is inherent in bodies2 and impressed is exterior to them.3 From the innate kind, there is only one force, called force of inertia.4 All the others, like pressure or impact, are impressed forces. Force of inertia justifies that a body resists change of its motion or resting. Changes in motion require impressed forces. With these two kinds of force the axioms are connected. According to the first law of motion, a body perseveres in its state of resting or of moving uniformly in a straight line, unless an impressed force constrains it to change its state.5 The second law of motion states: ‘‘The change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed’’.6 By ‘‘motion’’ is understood ‘quantity of motion’, i.e., the product of mass and velocity of the body (Definition II). A force is double another one, Newton adds, if the

1

‘‘Est autem vis impressa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta’’ (1726, p. 2).

2

‘‘Definitio III. Materiae vis insita est potentia resistendi, qua corpus unumquodque, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum ‘‘(p. 2).

3

‘‘Definitio IV. Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi vel movendi uniformiter in directum. Consistit haec vis in actione sola, neque post actionem permanet in corpore. Perseverat enim corpus in statu omni novo per solam vim inertiae’’ (p. 2).

4

‘‘Per vim insitam intelligo solam vim inertiae’’ (p. 389).

5

‘‘Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare’’ (p. 13).

6

‘‘Lex II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur ‘‘(p. 13).

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change caused by the first is double the change caused by the second.7 Force is, therefore, the concept for changes in the quantity of motion. From a formal point of view, force is a deviation from the motion referred to in the first law. As this motion is ascribed to a body on its own, force is consequently an external action. Euler’s Mechanics or the Science of Motion Presented Analytically, (1736), consists of two books: the first deals with free motions and the second with constrained motions. Euler’s approach to free motion is based on the following sequence: a body by itself stays at rest or maintains the uniformity and rectilinearity of motion.8 Force is that which changes these states.9 In conformity with this, Euler carries out the decomposition of force. If a body is moving on a plane, two components of force are considered:10 tangential force, whose effect is only the change of velocity,11 and radial force, which has no other effect than the change of direction of the motion.12 Euler’s approach can be interpreted in the following way. The mechanical states of a body by itself correspond to the motion of reference. Force is a deviation from this motion. The components of force correspond to the negations of the characteristics of the motion of reference. Let us move on to constrained motion. Euler was the first to deal with the motion constrained by a surface. In this case, three components of force are considered. The first one concerns the pressure exerted by a surface upon a moving body. This component must exist if the motion is conditioned. The other two components can exist or not. If they do not exist, a body covers the shortest line on the surface and moves uniformly, says Euler. If a body does not move uniformly, then there is a component tangential to the motion. If the body does not cover the shortest line, another component is considered.13 Euler’s approach can be interpreted as follows. The motion of a body constrained by a surface—it covers the shortest line uniformly—represents the motion of reference under those circumstances. Force is a deviation from that motion. The components of force correspond to the negation of the characteristics of this motion of reference. A difficulty arises concerning the connection of the concept of force with the phenomena. Euler deals briefly with this topic in the scholium to the definition of force (§ 102) and in another scholium, in the second book, in the following way. It is difficult to think of 7

‘‘Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, sive simul & semel, sive gradatim & successive impressa fuerit’’ (p. 13).

8

‘‘Corpus absolute quiescens perpetuo in quiete perseverare debet, nisi a causa externa ad motum sollicitetur’’ (Vol. I, § 56). ‘‘Corpus absolutum habens motum aequabiliter perpetuo movebitur, et eadem celeritate iam antea quovis tempore fuit motum, nisi causa externa in id agat aut egerit’’ (Vol. I, § 63). ‘‘Corpus absoluto motu praeditum progredietur in linea recta, seu spatium, quod describit, erit linea recta’’ (Vol. I, § 65).

9

‘‘Potentia est vis corpus vel ex quiete in motum perducens vel motum eius alterans’’ (vol. I, § 99).

10

‘‘Si corpus in eodem plano moveatur in eoque etiam positae sint potentiarum sollicitantium directiones, singulae potentiae resolvi possunt in binas, quarum altera sit normalis, altera tangentialis’’ (Vol. I, § 550). 11

‘‘Vis igitur tangentialis in corpus, dum elementum Mm percurrit, alium effectum non exerit, nisi quod motum eius vel acceleret vel retardet’’ (Vol. I, § 544).

12 ‘‘In hoc vero eius effectus consistit […] ut corporis tantum directionem immutet et efficiat, ut corpus, quod per se in recta esset progressurum, in linea curva promoveatur’’ (Vol. I, § 549). 13 ‘‘Prima potentia M, cuius directio in superficiem est normalis, nullum habebit effectum in immutando corporis motu, sed tota impendetur in pressionem superficiei. […] Secunda potentia N, quia eius directio in ipsa superficie est posita et normalis in directionem corporis, corporis directionem tantum immutabit celeritatem neque augendo neque minuendo. Haec vis igitur corpus a linea brevissima deducet facietque, ut non amplius in plano ad superficiem normali moveatur […]. Tertia potentia T, quia in directione corporis est posita, celeritatem tantum vel auget vel diminuit’’ (Vol. II, § 79).

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force without motion. Otherwise, motion can exist without force. Hence, he concludes that all forces which we observe, have origin in motions.14 This difficulty with observing force became the problem of the concept. Some theories of mechanics were carried out in order to avoid that concept of force. The first of them is dealt with next. In 1743, d’Alembert published the Traite´ de Dynamique, whose main contribution is a general method of solving problems. The second part of the book, which consists of two parts, is dedicated to the method and its applications. The first deals with the principles of mechanics which justify the method: the principles of inertia, equilibrium and the composition of motion.15 The first principle states that a body maintains its rest or if moving, will move rectilinearly and uniformly, if no external causes act on it.16 As accelerated motions are observable, there can be no doubt concerning the existence of those causes.17 There was, however, an objection against the use of force in mechanics. The cause of motion was represented at that time by u in the equation udt = du, where dt and du represent small quantities of time and velocity. D’Alembert includes this equation in his theory: it defines accelerative force.18 However, he does not accept its meaning.19 According to him, the thesis that force is the cause of acceleration is based on the ‘‘vague’’ and ‘‘obscure’’ principle that the cause is proportional to the effect.20 In fact, he continues, excepting impact, force is unknown to us.21 It is said, he exemplifies, that weight is the cause of acceleration by falling. However, what is observed is only the motion and not the force.22 Hence, he carried out a theory of mechanics without supposing 14 ‘‘Motum enim semel existentem perpetuo conservari debere clare ostendimus supra (§ 63); hic vero, quemadmodum ex motu potentiae oriantur, exposuimus. Quemadmodum vero potentiae sine motu vel existere vel conservari queant, concipi non potest. Quamobrem concludimus omnes potentias, quae in mundo conspiciuntur, a motu provenire’’ (Vol. II, § 29). 15 ‘‘Le Principe de l’e´quilibre joint a` ceux de la force d’inertie & du Mouvement compose´, nous conduit a` la solution de tous les Probleˆmes […]’’ (1758, p. xv). 16 ‘‘Un Corps en repos y persistera, a` moins qu’une cause e´trangere ne l’en tire’’ (p. 3–4). ‘‘Un Corps mis une fois en mouvement par une cause quelconque, doit y persister toujours uniforme´ment & en ligne droite, tant qu’une nouvelle cause, diffe´rente de celle qui l’a mis en mouvement, n’agira pas sur lui’’ (p. 4). 17 ‘‘On appelle en ge´ne´ral puissance ou cause motrice, tout ce qui oblige un Corps a` se mouvoir’’ (p. 4). ‘‘Cette variation continuelle ne peut provenir (art. 6.) que de quelque cause e´trangere qui agit sans cesse, pour acce´le´rer ou retarder le Mouvement’’ (p. 17). 18 ‘‘nous nous contenterons […] d’entendre seulement par le mot de force acce´le´ratrice, la quantite´ a` laquelle l’accroissement de la vitesse est proportionnel’’ (p. 25). 19 ‘‘La pluˆpart des Ge´ometres pre´sentent sous un autre point de vuˆe l’e´quation udt = du entre les tems & les vitesses. Ce qui n’est, selons nous, qu’une hypothese, est e´rige´ par eux en principe. Comme l’accroissement de la vitesse est l’effet de la cause acce´le´ratrice, & qu’un effet, selon eux, doit eˆtre toujours proportionnel a` sa cause, ces Ge´ometres ne regardent pas seulement la quantite´u comme la simple expression du rapport de du a` dt; c’est de plus, selon eux, l’expression de la force acce´le´ratrice, a` laquelle ils pre´tendent que du doit eˆtre proportionnel, dt e´tant constant’’ (p. 24–25). 20 ‘‘Pourquoi donc aurions-nous recours a` ce principe dont tout le monde fait usage aujourd’hui, que la force acce´le´ratrice ou retardatrice est proportionnelle a` l’e´le´ment de la vitesse? principe appuye´ sur cet unique axiome vague & obscur, que l’effet est proportionnel a` sa cause. […] nous nous contenterons d’observer, que […] il est inutile a` la Me´chanique, & que par conse´quent il doit en eˆtre banni’’ (p. xii). 21 ‘‘Le Mouvement uniforme d’un Corps ne peut eˆtre alte´re´ que par quelque cause e´trangere. Or de toutes les causes, soit occasionnelles, soit imme´diates, qui influent dans le Mouvement des corps, il n’y a tout au plus que l’impulsion seule dont nous soyons en e´tat de de´terminer l’effet par la seule connoissance de la cause, comme on le verra dans la seconde Partie de cet Ouvrage. Toutes les autreˆs causes nous sont entie´rement inconnues’’ (p. 22). 22 ‘‘toutes les autres causes ne se font connoıˆtre que par l’effet, & nous en ignorons entie´rement la nature: telle est la cause qui fait tomber les Corps pesans vers le centre de la Terre’’ (p. xi).

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a knowledge of the nature of force. He did not presume, for instance, that forces act together as each of them would act by itself.23 He further argues that the knowledge of motion is enough for science and opted to work out a theory starting from matter and motion.24 In sum, d’Alembert admitted force from an ontological point of view but not as an object of knowledge due to the lack of observability. Euler in 1750, published an article with the title ‘‘Discovery of a New Principle of Mechanics’’. The new principle is only an equation, whose form is ‘force = mass
acceleration’.25 Force, which he also called ‘‘potentia’’ (1736), is decomposed into three components, symbolized by Px, Py, and Pz. The components have the form Px = 2 m d2x/dt2.26 With this kind of decomposition, the information concerning the path covered by a body and how it is covered lies in the coordinates and in the acceleration along the coordinates. Lagrange’s Analytical Mechanics, 1788, made a new contribution to the concept of force and to the decomposition of motion. This book consists of two parts: statics and dynamics. Statics is defined as the science of the equilibrium of forces27 and dynamics as the science of forces and of the motions which are caused by them.28 The concept of force is presented at the beginning of the first part, as the cause or tendency to cause motion in a body.29 This meaning will change in the course of the development of the theory. This development is connected with the proposition which unifies the theory, and will be considered briefly: the addition of the moments of force equals zero. In statics, two forces are in equilibrium if their values, P and Q, and distances to the fulcrum, dp and dq, are related as follows: P dp = -Q dq. The general equation for the 23 ‘‘Quelques Lecteurs pourront eˆtre surpris de ce que je tire la de´monstration d’une proposition si simple en apparence, d’un cas ge´ne´ral beaucoup plus compose´; mais on ne peut, ce me semble, de´montrer autrement la proposition dont il s’agit ici, qu’en regardant comme un axiome incontestable, que l’effet de deux causes conjointes est e´gal a` la somme de leurs effets pris se´pare´ment, ou que deux causes agissent conjointement comme elles agiroient se´pare´ment; principe qui ne me paroıˆt pas assez e´vident, ni assez simple, qui tient d’ailleurs de trop pre`s a` la question des forces vives, & au principe des forces acce´le´ratrices dont nous avons parle´ ci-dessus art. 22. C’est la raison qui m’a oblige´ a` e´viter d’en faire usage’’ (p. 38–39). 24 ‘‘A l’e´gard des de´monstrations de ces Principes en eux-meˆmes, le plan que j’ai suivi pour leur donner toute la clarte´ & la simplicite´ dont elles m’ont paru susceptibles, a e´te´ de les de´duire toujours de la conside´ration seule du Mouvement, envisage´ de la maniere la plus simple & la plus claire. Tout ce que nous voyons bien distinctement dans le Mouvement d’un Corps, c’est qu’il parcourt un certain espace, & qu’il employe un certain tems a` le parcourir. C’est donc de cette seule ide´e qu’on doit tirer tous les Principes de la Me´chanique, quand on veut les de´montrer d’une maniere nette & pre´cise’’ (p. xvi). ‘‘De toutes ces re´flexions, il s’ensuit que les loix de la Statique & de la Me´chanique, expose´es dans ce Livre, sont celles qui re´sultent de l’existence de la matiere & du mouvement’’ (p. xxviii). 25 ‘F = ma’ is usually called ‘Newton’s second law’. According to Euler, who read the Principia, Newton’s second axiom is not expressed by that equation. As a matter of fact, that equation does not appear anywhere in Newton’s Principia. Moreover, historians of science do not agree with each other concerning an equation for Newton’s second law. (See Cohen 1970, p. 144; Dellian 1985, p. 401; Maltese 1992, p. 26). 26 Euler writes the equation firstly to coordinate x and afterwards similarly for y and z. ‘‘Apre`s l’e´le´ment du tems dt, soit x ? dx la distance du corps au plan et prenant cet e´le´ment dt pour constant, il sera 2Mddx ¼ P dt2 ; selon que la force P tend ou a` e´loigner ou a` approcher le corps du plan. Et c’est cette formule seule, qui renferme tous les principes de la Me´canique’’ (Opera Omnia, Ser. II, Vol. 5, p. 89). 27 ‘‘La Statique est la science de l’e´quilibre des forces’’ (1888–1889, Vol. I, p. 1). 28 ‘‘La Dynamique est la science des forces acce´le´ratrices ou retardatrices et des mouvements varie´s qu’elles doivent produire’’ (Vol. I, p. 237). 29 ‘‘On entend, en ge´ne´ral, par force ou puissance la cause, quelle qu’elle soit, qui imprime ou tend a` imprimir du mouvement au corps auquel on la suppose applique´e’’ (Vol. I, p. 1).

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equilibrium of two forces can, therefore, be written in the form P dp ? Q dq = 0. P dp is called ‘moment of force’ and Q dq as well.30 Thus, that equation can be read: the sum of the moments of forces is equal to zero. Equations with three or more forces in statics as well as the equations of motion can be read in the same way. The general equation of dynamics in generalised coordinates, for instance, is written in the form Ndn þ Wdw þ . . . ¼ 0 where ‘‘Ndn’’ is a ‘moment of force’. The elements of ‘moment of force’ have been generalised.31 This generalisation might obscure the meaning of force. Let us consider a simple example to clarify the interpretation: the circular movement of a point around an axle. Supposing that the motion takes place on the plane XY and taking the angle h between the X-axis and the radius as the generalised coordinate, a short piece of the path ds is given by r dh. Once the geometrical element of a motion is determined, Lagrange’s equations indicate how the path is covered. The two possible forms are h¼0 mr2 € or h ¼ Qh : mr 2 € In the first case, the motion is uniform; in the second, there is a deviation from uniformity. From a formal point of view, the circular uniform motion functions here as the motion of reference. There is place for force only if that motion changes. Lazare Carnot developed a new theory of mechanics in order to avoid the concept of force as the cause of acceleration. According to his Principes fondamentaux du mouvement et du repos, (1803), there are two ways of carrying out mechanics: either as a theory of force or as a theory of motion.32 The first one was followed by almost all the authors, said Carnot. He also acknowledged its advantages. It has, however, one shortcoming, being based on the ‘‘metaphysical’’ concept of force. This gave him the reason for opting for the second method.33 The problem pointed out by Carnot concerns the observation of force

30 ‘‘Nous nommerons chaque terme de cette formule, tel que Pdp, le moment de la force P […] la formule ge´ne´rale de la Statique consistera dans l’e´galite´ a` ze´ro de la somme des moments de toutes les forces’’ (Vol. I, p. 29–30). 31 Some examples of generalization. 1.’’Nommons E la force de l’e´lasticite´ et e l’angle exte´rieur qu’elle tend a` diminuer; le moment de cette force sera exprime´ par Ede (Sect. II, art. 9), de sorte que la somme des moments de toutes les forces du syste`me sera […] ? E de.’’ (Vol. I, p. 143). 2. ‘‘Appliquons les meˆmes principes a` la de´termination de l’e´quilibre d’une surface dont tous les e´le´ments dm soient extensibles et contractibles […]’’ (See Vol. I, p. 158–159). 3. ‘‘A l’e´gard de la quantite´ k dont nous venons de de´terminer la valeur, il est bon de remarquer que le terme Sk dL de l’e´quation ge´ne´rale de l’article 10 repre´sente la somme des moments d’autant de forces k qui tendent a` diminuer la valeur de la fonction L […]’’ (See Vol. I, p. 214–215). 32 ‘‘Il y a deux manie`res d’envisager la me´canique dans ses principes. La premie`re est de la conside´rer comme la the´orie des forces, c’est-a`-dire des causes qui impriment les mouvemens. La seconde est de la conside´rer comme la the´orie des mouvemens eux-meˆmes’’ (p. xi). 33 ‘‘La premie`re me´thode offre donc beaucoup plus de facilite´; aussi est-elle, comme je l’ai observe´ cidessus, presque ge´ne´ralement suivie’’ (p. xv–xvi). ‘‘La premie`re est presque ge´ne´ralement suivie, comme la plus simple; mais elle a le de´savantage d’eˆtre fonde´e sur une notion me´taphysique et obscure qui est celle des forces’’ (p. xi–xii). ‘‘j’ai adopte´ ici la seconde comme je l’avois de´ja` fait dans la premie`re e´dition; parce que j’ai voulu e´viter la notion me´taphysique des forces’’ (p. xvi).

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and was presented in considering machines, according to the author, as the most important object of mechanics. Some machines from that time worked thanks to men or animals. If a human being brings a machine into motion, he is the cause of that motion. The cause of motion was force, according to science. Is this force, Carnot questioned, the structure of the skeleton of the human being or of an animal or their wills? Does a double force mean, he continued to ask, that the will in the first case is double that in the other?34 This questioning shows the difficulty.35 As a solution to this problem, Carnot proposed to identify force with the quantity of motion which a force caused in a body.36 In doing this, we do not know more about the force which causes motion but it does not disturb the theory.37 Within this, force is a certain quantity of motion or, in other words, the motion caused by real force, called ‘‘first cause’’. In the introduction to the book, Carnot defends the thesis that what we know comes from experiments.38 From them, Carnot drew 7 statements called hypotheses, which constitute the starting point of his theory.39 The first of them corresponds to the law of inertia.40 Once admitted that a body by itself maintains its resting or moving rectilinearly and uniformly, it follows that whatever motion requires an external cause. The ‘‘first cause’’ satisfies this requirement. It does not satisfy, however, the epistemological requirement of observation. Barre´ de Saint-Venant also carried out a reorganization of mechanics in Principles of Mechanics, (1851). This book is divided into three thematic domains: kinematics, dynamics and statics. In kinematics, the motion is considered ‘‘merely geometric’’. It is 34 ‘‘Ces causes sont-elles la volonte´ ou la constitution physique de l’homme ou de l’animal qui par son action fait naıˆtre le mouvement? Mais qu’est-ce qu’une volonte´ double ou triple d’une autre volonte´, ou une constitution physique capable d’un effet double ou triple d’une autre?’’ (p. xii). 35 ‘‘quelle ide´e nette peut pre´senter a` l’esprit en pareille matie`re le nom de cause? il y a tant d’espe`ces de causes! Et que peut-on entendre dans le langage pre´cis des mathe´matiques par une force, c’est-a`-dire, par une cause double ou triple d’une autre?’’ (p. xii). 36 ‘‘Si l’on prend le parti de ne point distinguer la cause de l’effet, c’est-a`-dire, si l’on entend par le mot force la quantite´ de mouvement meˆme qu’elle fait naıˆtre dans le mobile auquel elle est applique´e, on devient intelligible’’. (p. xii-xiii). ‘‘Je re´pe´terai d’abord, qu’il ne s’agit point ici des causes premie`res qui font naıˆtre le mouvement dans les corps, mais seulement du mouvement de´ja` produit et inhe´rent a` chacun d’eux. C’est cette quantite´ de mouvement de´ja` produite dans un corps, qu’on nomme sa force ou sa puissance’’ (p. 47). ‘‘ainsi que nous l’avons de´ja` observe´, on ne conside`re, en me´canique, aucune force qui ne re´side effectivement dans les corps, c’est-a`-dire, qui ne soit re´ellement une quantite´ de mouvement de´ja` produite’’ (p. 108). 37 ‘‘La me´canique ne remonte pas jusqu’aux causes premie`res qui produisent le mouvement; elle n’examine pas comment la volonte´ de l’homme ou de l’animal fait sortir ses membres du repos, ou les y rame`ne spontane´ment: elle ne voit que le fait qui en re´sulte, ne conside`re que le mouvement de´ja` produit, et son objet est uniquement de rechercher comment ce mouvement une fois imprime´, se conserve, se propage ou se modifie’’ (p. 33). 38 ‘‘Les anciens e´tablirent en axioˆme que toutes nos ide´es viennent des sens: et cette grande ve´rite´ n’est plus aujourd’hui un sujet de contestation. Il suit de-la`, que toute science quelconque tire ses e´le´mens de l’expe´rience, puisque les premie`res ide´es qu’elle puisse combiner sont le re´sultat de nos sensations, qui ne sont autre chose que les donne´es de l’expe´rience. ‘‘D’ou` l’homme tire-t-il, dit Locke, tous ces mate´riaux qui sont comme le fond de tous ses raisonnemens et de toutes ses connoissances? Je re´ponds en un mot, de l’expe´rience’’ (p. 2). 39 ‘‘On pourra remarquer que ces hypothe`ses rentrent en partie les unes dans les autreˆs: mon objet n’a pas e´te´ de les re´duire au plus petit nombre possible; il me suffit qu’elles ne soient point contradictoires et qu’elles soient clairement entendues’’ (p. 47). 40 ‘‘Cette hypothe`se est le principe connu sous le nom de loi d’inertie’’ (p. 53).

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considered from a physical point of view in dynamics, according the author. Statics is presented as a special case of dynamics. His dynamics presents a new sequence of the concepts of force, mass, and acceleration. Barre´’s theory starts from a unique proposition, which states that the acceleration of bodies depends on the points which constitute them. The number of points is considered proportional to the mass of a body.41 The mass of a body is, however, not determinable by those points. For the measurement of mass, the reciprocal alteration of the velocities of bodies by impact is proposed.42 As the use of this measurement process is very difficult, Saint-Venant recommended measuring mass by weighing.43 Force is defined as the product of mass by acceleration.44 At the beginning of the Principles and again in dynamics, the problems concerning force as cause of acceleration are referred to. The author aims to overcome those difficulties in making force a mere mathematical concept. However, other difficulties arose: not only concerning the process of measurement of mass but also the interpretation of phenomena. The terms used in the dealing with phenomena remind us of the traditional concept of force: ‘acting forces’, ‘force acts on bodies’, and analogous expressions.45 Saint-Venant was aware of this: he indicated at the beginning of dynamics, how the traditional interpretation of phenomena is to be understood in compliance with the planned conceptual framework.46 In sum, Barre´’s aim was to make of force a mathematical concept due to the problems caused by the concept in mechanics. For this reason, he starts with acceleration and defines mass through the impact of two bodies. This measurement process as well the interpretation of phenomena caused difficulties. 41

‘‘On donne le nom de Masses a` des nombres proportionnels a` ceux des points e´le´mentaires qu’il faut supposer dans les corps, comparativement les uns aux autreˆs, pour expliquer leurs divers mouvemens par cette loi [la loi ge´ne´rale], conforme´ment a` son e´nonce´’’ (§ 81). 42 ‘‘La masse d’un corps est le rapport de deux nombres exprimant combien de fois ce corps et un autre corps choisi arbitrairement et constamment le meˆme, contiennent de parties qui, e´tant se´pare´es et heurte´es deux a` deux l’une contre l’autre se communiquent, par le choc, des vitesses oppose´es e´gales’’ (§ 81). 43 ‘‘Mais on peut, en ge´ne´ral, se dispenser de ces mesurages de vitesse et d’acce´le´ration, qui sont de´licates et difficiles, et estimer promptement les masses […] par le pesage’’ (§ 88). ‘‘Les poids des corps sont, comme l’on voit, en un meˆme lieu, proportionells aux masses’’ (§ 89). 44 ‘‘La force ou l’action attractive ou re´pulsive d’un corps sur un autre est une ligne ayant pour grandeur le produit de la masse de celui ci par l’acce´le´ration moyenne de ses points vers ceux du premier et pour direction celle de cette acce´le´ration’’ (§ 81). 45 See §§ 83, 85, 86, 93, 97, 98, 100, 103, 109, 116, 119, 120, 129, 138, 145, 157, 159, 161, 163, 164, 166, 167, 168, 171, 172, 173, 174, 175, 178, 179, 184, 185. Some examples:—‘‘l’acce´le´ration g qu’ils [les poids] donnent aux masses sur lesquelles ils agissent’’ (§ 93);—‘‘Si les forces agissant sur le systeˆme se font e´quilibre […]’’ (§ 119). ‘‘Force’’ or ‘‘puissance’’ appear also with the verbs:—‘‘solliciter’’, §§ 97, 120, 159, 169, 172;—‘‘appliquer’’, §§ 168, 169, 185;—‘‘exercer’’, §§ 144, 179, 185. 46 ‘‘La de´nomination de force ou d’action vient du sentiment de l’effort que nous exerc¸ons lorsque nous voulons imprimer une acce´le´ration a` un corps et de ce que, dans le langage commun, l’on attribue me´taphoriquement une activite´ analogue a` celle de l’homme, aux autreˆs eˆtreˆs, meˆme inanime´s, dans la direction desquels l’on voit des corps prendre un mouvement. Pour nous conformer a` cette manie`re de parler qui a passe´ dans la science, nous dirons quelque fois qu’un corps A est sollicite´ par une force de grandeur F, e´manant d’un autre corps B, et qui, en agissant sur A dans une certaine direction, produit une acce´le´ration j ou donne a` A une vitesse jt dans le temps t. Mais, par la`, nous voudrons dire simplement que les points du corps A ont, vers ceux du corps B, des composantes d’acce´le´ration dont la moyenne a une certaine direction et une grandeur qui, multiplie´e par la masse m de A, donne un produit mj e´gal a` F. Nous dirons que nous appliquons une force F a` un corps A dans une certaine direction: cela signifiera que nous plac¸ons un ou plusieurs autreˆs corps anime´s ou inanime´s dans des situations ou dans un e´tat physique tels que les acce´le´rations des points de A vers leurs points aient une moyenne qui, multiplie´e par la masse de A, donne F’’ (§ 82).

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In the following year, Reech published the Cours de Me´canique, whose aim was, however, very different. This book consists of two parts: first, ‘on the alleged science of mechanics’ and second, ‘on the true science of mechanics’. The reason for such a distinction lies in the concept of force. Criticised by him is the ‘relative concept of force’. By this is understood the measurement of force through its ‘evident geometric effect’, i.e., by the deviation from a certain motion. Instead, Reech proposes the ‘real, absolute concept of force’. This reality is connected with the sensation of our muscles. Our sensation awakened in us the idea of a certain quantity, called pressure or traction, which is the cause of the alteration of the motions of bodies which have been touched. This is the true idea of force that we should have, says Reech in the introduction to the Course.47 In the corpus of the book, force is defined as pressures or tractions that we can make through our organs on the bodies surrounding us.48 The process of measurement proposed for forces reflects the definition. Force is to be measured through a convenient thread, whose changes in length indicate the magnitude of force.49 In this process, some difficulties lie inherently. Reech pointed out that a thread has some mass and this influenced the measurement of force. He proposed then, making a conceptual distinction between matter and connections of material points and to ascribe mass only to matter. In doing so, the thread is considered massless.50 Another difficulty concerns the limitations in using a thread to measure force, as, for instance, in the case of celestial motions. For such cases, he proposes measuring force thanks to the deviation from a certain, conventional motion. To play this role, he chooses the rectilinear and uniform motion, not because it is the natural one, he points out, but only because it is the simplest motion and the most commonly used one. The law of inertia is, according to him, a mere convention.51 In 1876, Kirchhoff published a textbook on Mechanics, which became very successful; the second edition occurred in the same year. The preface to the book announces a restructuring of mechanics, whose leitmotiv lies in the concept of force. Physicists 47

‘‘La seule et ve´ritable ide´e que nous devions nous faire de la force, c’est celle que nous acque´rons quand, a` l’aide de nos organes, nous cherchons a` modifier l’e´tat de repos ou de mouvement des corps qui nous environnent. Nous e´prouvons alors des sensations qui e´veillent en nous plusieurs ide´es fondamentales: d’abord celle de l’existence des corps, puis celle de la forme des corps et des proprie´te´s de l’espace, puis celle du mouvement et du temps, puis encore celle d’une certaine quantite´ que nous nommons une pression ou une traction. Cette quantite´ est une cause de mouvement ou plutoˆt une cause de changement de mouvement pour les parties des corps que nous rencontrons a` l’aide de nos organes’’ (p. 37). 48 ‘‘Par le mot force, on ne doit entendre que les pressions ou tractions que nous pouvons faire a` l’aide de nous organes, sur les corps qui nos environnent’’ (p. 57). 49 ‘‘La direction de la force sera celle du fil dans lequel elle re´sidera, et l’intensite´ de la force de´pendra de l’allongement ainsi que de la nature du fil’’ (p. 46). 50 ‘‘Par une abstraction de notre entendement, nous pouvons nous repre´senter un fil tendu, comme e´tant comple´tement de´pourvu de sa qualite´ matie`re ou masse, et alors un pareil fil sera parfaitement indiffe´rent a` se mouvoir d’une manie`re plutoˆt que d’une autre, c’est-a`-dire qu’un pareil fil suivra spontane´ment les corps ou obstacles, auxquels il se trouvera attache´, en faisant de la force aux points d’attache sur ces obstacles, et en n’exigeant aucune force pour participer a` leur mouvement’’ (p. 59). 51 ‘‘Mais alors, il y aura une convention a` faire. Il s’agira de savoir quelle sorte de mouvement, rectiligne ou curviligne, uniforme ou varie´, nous devrons admettre, comme e´tant celui d’un point mate´riel entie`rement libre en apparence, et parce que nous aurons une entie`re latitude a` cet e´gard, ainsi que nous l’avons de´ja` fait pressentir dans la dernie`re section de la premie`re partie, avec le seul avantage ou inconve´nient d’en voir re´sulter de plus ou moins grandes simplifications dans les relations me´caniques des syste`mes, nous serons conduits naturellement a` faire servir a` un tel usage l’e´tat de mouvement rectiligne uniforme, et a` rencontrer cette fameuse loi d’inertie de la matie`re, qui ne sera plus un principe ni un fait d’expe´rience, mais une pure convention, la plus simple de toutes celles parmi lesquelles nous nous trouverons oblige´s de choisir’’ (p. 49).

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disagreed with each other, according to Kirchhoff, about if some important statements, such as if the law of inertia and the parallelogram of forces were axioms, theorems or experimental results. According to the author, these problems lie in the concept of force: in the lack of clarity of ‘cause’ motion and ‘tendency to cause’ motion.52 To avoid these problems, Kirchhoff restricts the function of the science to the description of motion.53 At the beginning of the book, mechanics is defined as the science of motion. As in order to conceive motion, the notions of space, time and matter are necessary and sufficient, according to the author, these become the primitive notions of the science of motion. Force and mass are to be constructed within the theory.54 This plan was, however, not carried out successfully. Kirchhoff himself detected the following difficulty. If a system of forces acts on a body, it is impossible to determine that system only through that motion. Thanks to the observation of that motion, we achieve the resultant but not the components of force. There are, therefore, forces which cannot be subsumed by the theory.55 Another difficulty concerns the interpretation of phenomena. The most common expressions connected with ‘force’ are: ‘forces act’, ‘acting forces’, ‘forces are exerted’, ‘exerting forces’.56 If forces act, they must be something which have an influence. Thus, this terminology leads us to think of force as a cause, which Kirchhoff had planned to avoid. In sum, Kirchhoff changed the status of force, from a real thing to a mere theoretical concept. There was, however, a difficulty in carrying out that transformation, namely in obtaining every force through motion and in the interpretation of phenomena. Mach’s Mechanics was published in 1883, with successive editions and reprints. His solution for force was taken up from a short paper written in 1868. Here, he criticized the vicious circle in defining mass at that time: weight was defined by mass and mass by weight.57 He proposed, then, a solution which presents the sequence: acceleration, mass, force. The starting proposition of Mach’s proposal, presented anew in Mechanics (1933), says that bodies in interaction cause reciprocal acceleration.58 This is considered a matter of 52 ‘‘Man pflegt die Mechanik als die Wissenschaft von den Kra¨ ften zu definiren, und die Kra¨fte als die Ursachen, welche Bewegungen hervorbringen oder hervorzubringen streben. Gewiss ist diese Definition […] Aber ihr haftet die Unklarheit an, von der die Begriffe der Ursache und des Strebens sich nicht befreien lassen. Diese Unklarheit hat sich z. B. gezeigt in der Verschiedenheit der Ansichten daru¨ber, ob der Satz von der Tra¨gheit und der Satz vom Parallelogramm der Kra¨fte anzusehen sind als Resultate der Erfahrung, als Axiome oder als Sa¨tze, die logisch bewiesen werden ko¨nnen und bewiesen werden mu¨ssen’’ (1897, p. V). 53 ‘‘Aus diesem Grunde stelle ich es als die Aufgabe der Mechanik hin, die in der Natur vor sich gehenden Bewegungen zu beschreiben, und zwar vollsta¨ndig und auf die einfachste Weise zu beschreiben’’ (p. V). 54 ‘‘Zur Auffassung einer Bewegung sind die Vorstellungen von Raum, Zeit und Materie no¨thig, aber auch hinreichend. Mit diesen Mitteln muss die Mechanik suchen, ihr Ziel zu erreichen, und mit ihnen muss sie die Hu¨lfsbegriffe construiren, die sie dabei no¨thig hat, z. B. die Begriffe der Kraft und der Masse’’ (p. 1). 55 ‘‘Es ist einleuchtend, dass, wenn man eine bestimmte Bewegung eines Punktes als bedingt durch mehrere Kra¨fte ansieht, diese nicht einzeln bestimmt sind; nur die Resultante ist bestimmt […] Aus der Bewegung allein kann die Mechanik nach unserer Auffassung die Definitionen der Begriffe scho¨pfen, mit denen sie es zu thun hat. Es folgt daraus, dass nach Einfu¨hrung von Kra¨ftesystemen an Stelle einfacher Kra¨fte die Mechanik ausser Stande ist, eine vollsta¨ndige Definition des Begriffs der Kraft zu geben’’ (p. 11). 56 See for instance, pp. 8, 13, 22, 23, 25, 30, 31, 33, 34, 35, 36, 38, 39, 45, 51, 56, 60, 62, 68, 86, 88, 89, 109, 110, 115, 126, 127, 128, 132, 144, 146, 150, 160, 164, 165, 170, 171, 233, 235, 236, 244, 247, 249, 290, 308, 348, 349, 352, 358, 369, 377, 385, 393, 404, 416, 418, 436, 455, 458. 57 ‘‘Man definirt gewo¨hnlich m = p/g und wiederum p = mg’’ (1868, p. 356). 58 ‘‘Die Definition [der Masse] beru¨cksichtigt lediglich die Tatsache, daß in Wechselbeziehung stehende Ko¨rper, ob sogenannte Fernwirkungen, starre oder elastische Verbindungen in Betracht kommen, aneinander Geschwindigkeitsa¨nderungen (Beschleunigungen) bestimmen. Mehr als dies braucht man nicht zu wissen, um mit voller Sicherheit und ohne Furcht, auf Sand zu bauen, definieren zu ko¨nnen’’ (1933, p. 261).

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fact. Taking one body as a unit, Mach continues, the mass of the other is measured through the proportion of the accelerations due to the interaction of both bodies.59 Force is then defined as the product of mass and acceleration.60 Mach’s aim was also to make of force a mere theoretical concept. How to think of force remained, however, a problem. Force is any circumstance of which the consequence is motion, says Mach.61 This leads to the idea that force is the cause of motion. In a second step, Mach made recourse to our sensation to understand force: the circumstances determinative of motion which are best known to us, are our volitional acts. Thus, continues Mach, our habit of representing circumstances determinative of motion as something akin to volitional acts arises.62 He knew that this was not scientific but he did not know of a better way: the attempts to set aside this conception as subjective and unscientific, he said, fail invariably.63 Even though Mach gave a new definition of force and proposed to understand it as a mere theoretical concept, the approach to phenomena was marked by the traditional interpretation. Hertz’s (1894) posthumous work, The Principles of Mechanics, was published. In the introduction to the book, his philosophy of science is presented. On it is based his mechanical theory. With this modus procedendi the author aims to overcome some problems of mechanics. Among the main difficulties of this science is the concept of force. If we swing a stone tied to a piece of string in a circle, exemplifies Hertz, we are conscious of exerting a force upon the stone. This agrees with the definition of force: force is independent of motion and the cause of it. Newton’s third law, he continues, requires, however, an opposing force to the force exerted by the hand upon the stone. Here the

59 ‘‘Ist uns aber einmal durch mechanische Erfahrung die Existenz eines besondern beschleunigungbestimmenden Merkmals der Ko¨rper nahegelegt, so steht nichts im Wege, willku¨rlich festzusetzen: Ko¨rper von gleicher Masse nennen wir solche, welche aufeinander wirkend sich gleiche entgegengesetzte Beschleunigungen erteilen. Hiermit haben wir nur ein tatsa¨chliches Verha¨ltnis benannt. Analog werden wir in dem allgemeinern Fall verfahren. Die Ko¨rper A und B ([…]) erhalten bei ihrer Gegenwirkung beziehungsweise die Beschleunigungen -u und ?u, wobei wir den Sinn derselben durch das Zeichen ersichtlich machen. Dann sagen wir, B hat die -u/u fache Masse von A. Nehmen wir den Vergleichsko¨rper A als Einheit an, so schreiben wir jenem Ko¨rper die Masse m zu, welcher A das mfache der Beschleunigung erteilt, die er in Gegenwirkung von A erha¨lt. Das Massenverha¨ltnis ist das negative umgekehrte Verha¨ltnis der Gegenbeschleunigungen’’ (1933, p. 211–212). 60 ‘‘Bewegende Kraft ist das Produkt aus dem Massenwert eines Ko¨rpers in die an demselben bestimmte Beschleunigung’’ (1933, p. 242). 61 ‘‘Die Kraft ist also ein bewegungbestimmender Umstand dessen Merkmale sich in folgender Art angeben lassen. Die Richtung der Kraft ist die Richtung der von der gegebenen Kraft allein bestimmten Bewegung. Der Angriffspunkt ist derjenige Punkt, dessen Bewegung auch unabha¨ngig von seinen Verbindungen bestimmt ist. Die Gro¨ße der Kraft ist das Gewicht, welches, nach der bestimmten Richtung (an einer Schnur) wirkend, an dem gegebenen Punkt angreifend, dieselbe Bewegung bestimmt oder dasselbe Gleichgewicht erha¨lt’’ (1933, p. 75). 62 ‘‘Diejenigen bewegungbestimmenden Umsta¨nde, die uns am besten bekannt sind, sind unsere eigenen Willensakte, die Innervationen. Bei den Bewegungen, welche wir selbst bestimmen, sowie bei jenen, zu welchen wir durch a¨ußere Umsta¨nde gezwungen sind, empfinden wir stets einen Druck. Dadurch stellt sich die Gewohnheit her, jeden bewegungbestimmenden Umstand als etwas einem Willensakt Verwandtes und als einen Druck vorzustellen’’ (1933, p. 74). 63 ‘‘Die Versuche, diese Vorstellung als subjektiv, animistisch, unwissenschaftlich zu beseitigen, mißglu¨cken uns immer. Es kann auch nicht nu¨tzlich sein, wenn man seinen eigenen natu¨rlichen Gedanken Gewalt antut und sich zu freiwilliger Armut derselben verdammt’’ (1933, p. 74).

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problem begins: ‘‘In our laws of motion, force was a cause of motion, and was present before motion. Can we, without confusing our ideas, suddenly begin to speak of forces which arise through motion, which are a consequence of motion?’’ (1899, p. 6). Since force is defined as the cause of motion and there is a force that is a consequence of motion, the theory is not logically permissible, according to Hertz. His own solution for force appears in the following context. According to Hertz’s philosophy of science, a physical theory is an image which we form of things.64 His image starts from one axiom, which is formulated for free systems. A non-free system is understood as a part of a free system,65 which is not completely known.66 Hertz imagined, then, connections between the sub-systems and expressed them mathematically. To verify if the consequences of the image are conform with the respective phenomena, measurements must be made. The methods of determining force indicated by Hertz were common in the mechanics of that time (§§ 542–544). However, between these measurement processes and the constructed concept of force there is only a mere correspondence.67 The meaning of force in the theory is connected with its axiom. Hertz’s theory is based on one unique axiom called fundamental law. This proposition, the only one drawn from experiments, according to the author, has the form of the law of inertia: ‘‘Every free system persists in its state of rest or of uniform motion along the straightest path’’.68 In Hertz’s mechanics, there is a force if the motion is not uniform or the curvature of the path is not a minimum.69 Thus, force is a deviation from the motion referred to in the fundamental law. In sum, Hertz’s solution for force consists of a separation between force in thought, which belongs to the image, and force in practice, which is a measurement process. From a formal point of view, force is a deviation from the motion of the fundamental law. Poincare´ (1897) wrote an article about Hertz’s mechanics, in which he asserts categorically ‘‘to say that force is the cause of acceleration is to do metaphysics’’.70 He defends instead that a concept of force should be worked out from its measurement process. Hence, he began to consider the definition of equal forces: two forces are said to be equal if they attain equilibrium or produce the same acceleration on the same mass. Poincare´ comments: we cannot connect and disconnect forces to or from bodies as horses to coaches or engines to carriages. It was said as well that two forces are equal if they balance with the same weight. Poincare´ pointed out that the weight depends on the place. Furthermore, Newton’s third law 64

‘‘Wir machen uns innere Scheinbilder oder Symbole der a¨ußeren Gegensta¨nde, und zwar machen wir sie von solcher Art, daß die denknotwendigen Folgen der Bilder stets wieder die Bilder seien von den naturnotwendigen Folgen der abgebildeten Gegensta¨nde’’ (p. 1). 65 ‘‘Nach unserer Auffassung ist jedes unfreie System Teil eines gro¨ßeren freien Systems’’ (§ 429). 66 ‘‘Indem wir einen Teil eines freien Systems als unfreies System behandeln, setzen wir voraus, daß das u¨brige System uns mehr oder weniger unbekannt ist’’ (§ 430). 67 ‘‘Durch Anwendung einer jeden dieser drei Methoden ko¨nnen auch die Kra¨fte aus Rechnungsgro¨ßen zu Gegensta¨nden der unmittelbaren Erfahrung gemacht werden, d.h. zu Zeichen fu¨r bestimmte Verbindungen sinnlicher Empfindungen und Wahrnehmungen’’ (§ 541). 68 ‘‘Jedes freie System beharrt in seinem Zustande der Ruhe oder der gleichfo¨rmigen Bewegung in einer geradesten Bahn’’ (§ 309). 69 See § 368 (the differential equations of the motion of a free system) and § 482 (the equations of motion of a system influenced by forces). 70 ‘‘Quand on dit que la force est la cause d’un mouvement, on fait de la me´taphysique, et cette de´finition, si on devait s’en contenter, serait absolument ste´rile. Pour qu’une de´finition puisse servir a` quelque chose, il faut qu’elle nous apprenne a` mesurer la force; cela suffit d’ailleurs, il n’est nullement ne´cessaire qu’elle nous apprenne ce que c’est que la force en soi, ni si elle est la cause ou l’effet du mouvement’’ (1897, p. 734).

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is employed, in such cases, as a definition and not an experimental law. Due to all these problems, Poincare´ moved on to the other possibility of defining force, thanks to mass. To this kind of sequence of definitions—acceleration, mass, force—Poincare´ raised some objections as well. One of them concerns the supposition that the acceleration of body A is caused by B, when it is caused not only by B, but also by that of C, D, E, etc.71 To determine the mass of A based on the acceleration produced by B, it would be necessary to separate A’s acceleration into its elements.72 This decomposition of the acceleration would, however, only be possible, according to Poincare´, if the hypothesis of central forces was admitted.73 As the hypothesis does not offer guarantees, Poincare´ goes on to another possibility, the determination of mass through the center of mass.74 The center of mass of a system, on which no exterior action is exercised, is characterized by a uniform and rectilinear motion. Thus, the values of masses of bodies could be combined in such a way that the center of mass might show such a motion. Since there is not a system without exterior action, the law of the movement of the center of mass would only be valid for the whole of the universe. This means that such a determination of mass would imply the observation of the movement of the center of the universe, which, Poincare´ concludes, is an absurdity.75 From his analysis of the processes of measurement of force and mass, he concludes that it is impossible to give a satisfactory idea of mass and force within classical mechanics.76

3 Definitions in Textbooks In a sample of about a hundred textbooks (Voigt 1901; Lenard 1936; Sommerfeld 1947; Schaefer 1962; Budo´ 1974; Hestenes 1987; Alonso and Finn 1992; Daniel 1997; Gerthsen 2006; Kuypers 2008, among others), it was verified that ‘force is the cause of acceleration’ is the most common definition of force.

71 ‘‘l’acce´le´ration de A n’est pas due seulement a` l’action de B, mais a` celle d’une foule d’autreˆs corps C, D […]’’ (1897, p. 735). 72 ‘‘il faut donc de´composer l’acce´le´ration de A en plusieurs composantes, et discerner quelle est celle de ces composantes qui est due a` l’action de B’’ (1897, p. 735). 73 ‘‘Cette de´composition serait encore possible, si nous admettions que l’action de C sur A s’ajoute simplement a` celle de B sur A, sans que la pre´sence du corps C modifie l’action de B sur A, ou que la pre´sence de B modifie l’action de C sur A; si nous admettions, par conse´quent, que deux corps quelconques s’attirent, que leur action mutuelle est dirige´e suivant la droite qui les joint et ne de´pend que de leur distance; si nous admettions, en un mot, l’hypothe`se des forces centrales’’ (1897, p. 735). 74 ‘‘Mais avons-nous le droit d’admettre l’hypothe`se des forces centrales? Cette hypothe`se est-elle rigoureusement exacte? Est-il certain qu’elle ne sera jamais contredite par l’expe´rience? Qui oserait l’affirmer? Et si nous devons abandonner cette hypothe`se, tout l’e´difice si laborieusement e´leve´ s’e´croulera. Nous n’avons plus le droit de parler de la composante de l’acce´le´ration de A qui est due a` l’action de B. Nous n’avons aucun moyen de la discerner de celle qui est due a` l’action de C ou d’un autre corps. La re`gle pour la mesure des masses devient inapplicable’’ (1897, p. 735–6). 75 ‘‘Mais il n’existe pas de syste`me soustrait a` toute action exte´rieure; toutes les parties de l’Univers subissent plus ou moins fortement l’action de toutes les autreˆs parties. La loi du mouvement du centre de gravite´ n’est rigoureusement vraie que si on l’applique a` l’Univers tout entier. Mais alors il faudrait, pour en tirer les valeurs des masses, observer le mouvement du centre de gravite´ de l’Univers. L’absurdite´ de cette conse´quence est manifeste; nous ne connaissons que des mouvements relatifs; le mouvement du centre de gravite´ de l’Univers restera pour nous une e´ternelle inconnue’’ (1897, p. 736). 76 ‘‘nous devons conclure, qu’avec le syste`me classique, il est impossible de donner de la force et de la masse une ide´e satisfaisante’’ (1897, p. 736).

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Webster, for instance, wrote (1904): ‘‘The property of persistence thus defined is called Inertia. This gives a criterion for finding whether a force is acting on a body or not […] Force is acting on a body when its motion is not uniform’’ (p. 21). In Feynman’s Lectures (1974), one reads: ‘‘The Second Law gave a specific way of determining how the velocity changes under different influences called forces’’; and ‘‘If an object is accelerating, some agency is at work’’ (§ 9–4). Wolfson and Pasachoff, Physics (1990), write: ‘‘Why are we so interested in knowing about forces? Because forces cause changes in motion’’ (p. 76). If force is the cause of a motion which a body could not have by itself, force must be a real existing thing. A contrary thesis has, however, been defended by some physicists. Hamel in Mechanics (1912), says: ‘‘Force itself, however, we do not define as cause of motion, force is a thing of thought and not a natural phenomenon’’.77 Platrier writes in Rational Mechanics, (1954): ‘‘In fact, force is only a human concept and we have no knowledge of the profound cause of motions’’.78 Ludwig, in the Introduction to the Foundations of Theoretical Physics, (1985), defends the thesis that the concept of force does not describe anything which exists in reality. In his own terminology, force does not belong to ‘‘real text’’.79 Some physicists defend a variant of the most common definition of force, in understanding force as the effort felt by the pulling or pushing of an object. Planck (1916), for instance, says that the cause of the motion is called force and ‘‘it corresponds to the effort, which we feel, if that same motion had been produced through our muscles instead of the bodies, which caused it’’.80 Nolting 2005, writes: ‘‘The concept of force can only be defined indirectly through its effects. If we want to modify the state of movement or the shape of a body, for example, using our muscles, then an effort will be necessary […] This effort is called force […] We observe everywhere in our environment changes in the states of motion of certain bodies […] We see their causes equally in forces, which in the same way as our muscles, act on the bodies’’.81 This kind of definition of force has its origin in Reech’s theory (1852). An important follower of his was Jules Andrade, who wrote (1898), ‘‘Certain spirits despise the common idea of force, as furthermore, they despise the notion of muscular force. This disdain does

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‘‘Die Kraft selbst aber definieren wir nicht als Ursache der Bewegung; denn die Kraft ist ein Gedankending und keine Naturerscheinung’’ (p. 56). 78 ‘‘En re´alite´ la force ([F = m.a]) n’est qu’une conception humaine et la cause profonde des mouvements nous est inconnue’’ (p. 112). 79 ‘‘Der physikalische Begriff der Kraft beschreibt eben nicht etwas unmittelbar Feststellbares […] Der Kraftbegriff geho¨rt nicht zur Formulierung der Abbildungsprinzipien, die etwas im Realtext, d. h. an der Wirklichkeit ([…]) Ablesbares in eine mathematische Form umzuschreiben gestatten’’ (p. 145). 80 ‘‘Wir bezeichnen also nun ganz allgemein bei jeder beliebigen Bewegung die Ursache der Bewegung als Kraft und setzen ihre Gro¨ße proportional der durch sie bewirkten Beschleunigung. Dieselbe entspricht derjenigen Anstrengung, die wir verspu¨ren wu¨rden, wenn wir die na¨mliche Bewegung, anstatt durch den betreffenden Ko¨rper, durch unsere Muskeln hervorrufen wu¨rden’’ (p. 10). 81 ‘‘Der physikalische Begriff der Kraft la¨ßt sich nur indirekt durch seine Wirkungen definieren. Wollen wir den Bewegungszustand oder die Gestalt eines Ko¨rpers z.B. durch Einsatz unserer Muskeln a¨ndern, so bedarf es einer Anstrengung, die um so gro¨ßer ist, je gro¨ßer die zeitliche Geschwindigkeitsa¨nderung (Beschleunigung) oder je sta¨rker die Deformation sein soll. Diese Anstrengung heißt Kraft. […] Nun beobachten wir ¨ nderungen in den Bewegungszusta¨nden gewisser Ko¨rper, ohne daß unsere u¨berall in unserer Umgebung A Muskeln direkten Einfluß ha¨tten. Ihre Ursache sehen wir ebenfalls in Kra¨ften, welche in gleicher Weise wie unsere Muskeln auf die Ko¨rper einwirken’’ (p. 109).

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not seem justified to me, since the only common notion of force is the fruitful notion; mechanics, we admit clearly, is essentially anthropomorphic’’.82 Poincare´ 1900, defended the thesis, however, that this notion of effort does not acquaint us with the true nature of force.83 He adds, the anthropomorphism cannot provide the foundation of anything truly scientific or philosophical.84 The fundamental equation of dynamics is sometimes used to define force. Fließbach 2007, for instance, writes as follows: ‘‘Newton’s second axiom embraces the following definitions and affirmations: 1. Definition of mass; 2. Definition of force 3. […]’’85 As the equation referred to as Newton’s second axiom (F = ma) is composed of three variables, the definition of mass will have to be given by force and acceleration. As force is defined by the same equation, it follows that it depends on what mass and acceleration are. As, however, what mass might be depends on force, we remain not knowing what both are. This kind of definition was criticized by Mach in 1868, as seen above (see Hestenes 1987, p. 590; de Lozano and Cardenas 2002, p. 596). Although the kinds of definitions of force considered above had already been criticized, the criticism is rarely taken into account by modern authors.86 Let us consider if the defining of force could be improved. 4 Philosophy Newton’s force represents a deviation from the states referred to in the law of inertia. Euler conceived force as a deviation from a certain motion in creating the theory of the motion constrained by a surface. Reech criticized force because it was considered as a deviation from a certain motion. In Hertz’s theory, there is force if there is a deviation from the motion of the fundamental law. This concept, force as a deviation from a certain motion, is also present in the decomposition of force. A ‘deviation’ from a certain motion corresponds to the ‘negation’ of this motion, from a logical point of view. If the motion has the properties ‘p and q’, the negation of this conjunction is equivalent to the disjunction of the negations ‘non-p or non-q’. If it is characterized as ‘rectilinear and uniform’, the negation is ‘non-rectilinear or non-uniform’. The components of force are those which make the motion non-rectilinear or nonuniform. These are therefore connected with the logical negation of the characteristics of the motion of reference. If such a motion is characterized by ‘the shortest line and 82 ‘‘Certains esprits me´prisent cette ide´e vulgaire de la force, comme ils me´prisent d’ailleurs la notion de l’effort musculaire. Ce me´pris ne me paraıˆt pas justifie´, car seule, la notion vulgaire de la force est la notion fe´conde; la me´canique, avouons-le hautement, est essentiellement anthropomorphique’’ (p. 138). 83 ‘‘cette notion d’effort ne nous fait pas connaıˆtre la ve´ritable nature de la force’’ (1900, p. 468). 84

‘‘L’Anthropomorphisme a joue´ un roˆle historique conside´rable dans la gene`se de la Me´canique; peut-eˆtre fournira-t-il encore quelquefois un symbol qui paraıˆtra commode a` quelques esprits; mais il ne peut rein fonder qui ait un caracte`re vraiment scientifique, ou un caracte`re vraiment philosophique’’ (1900, p. 468).

85 ‘‘Das 2. Newtonsche Axiom beinhaltet folgende Definitionen und Aussagen: 1. Definition der Masse. 2. Definition der Kraft. 3. […]’’ (p. 13–14). 86 French (1971, p. 170) is an exception to this: knowing the difficulties in defining force he does not give any definition.

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uniformity’, the components of force are those which make the trajectory ‘non-the shortest’ or the movement ‘non-uniform’. If the least curvature and uniformity characterize the motion of reference, there is force if the curvature of the path is not a minimum or the motion along it is not uniform. In this case, Hertz did not speak of components of force, since he used coordinates for the decomposition. With the introduction of coordinates, the information concerning the path and the motion along it, is given through these. If m€ xi ¼ 0 holds for each rectangular coordinate, the motion is rectilinear and uniform. If not, it is non-rectilinear or non-uniform. This holds mutatis mutandis for generalized coordinates. The example of the circular motion conh ¼ 0 the motion is uniform; if mr2 € h ¼ Qh 6¼ 0; it is not. (We sidered above shows: if mr 2 € say that Qh does not have the dimensions of force. This can now be understood in connection with the motion of reference. In the former case, the motion of reference is the rectilinear and uniform motion; in the latter, it is the circular uniform motion. As these motions differ from each other, their variations in respect to time differ as well.) This concept of force concerns a motion which is characterized by a trajectory and how it is covered. This is not yet the concept of force we use when we say: ‘body A exerts force f on body B’. It is true that the variable m appears in the equations referred to above. However, mass does not matter, since those equations hold for all mechanical bodies. This is however not the case concerning the left-hand side of F = ma, if F refers to the force of a body. The force f also comes from experiments but not in the same way, as we will see. What can be drawn from experiments concerning this issue was presented in a clear way by Helmholtz. In his Lectures on Theoretical Physics, (1911), he wrote: ‘‘Motion and acceleration are facts, which can be observed […] On the contrary, if one speaks of force as cause of this motion, one does not know more about the nature of force than can be gathered from the observation of the occurrence of the motion […] Therefore, nothing can be stated about force, which is not already known from acceleration’’.87 This can also be shown in the following way. Let us suppose that we have to study a motion about which we have no further information. To study this motion, we have to observe it carefully. The best means of achieving this goal are certainly stroboscopic images or filming. The result of this is some tens of images. Thanks to them, we can measure the piece of the path in each interval of time, determine the respective velocity and calculate the acceleration. This is all, however, we can draw from the data. We can say nothing about the force or mass of the moving body without further information. Acceleration is therefore the only one of the three magnitudes which can be drawn from a phenomenon. It follows that force requires more than one phenomenon. The proposition ‘A exerts force f on B’ requires therefore more than one phenomenon. In order to assert that the force of A is f, we have to carry out a set of experiments (Kohlrausch 1996, p. 133 ff; Arons 1990, p. 52 ff). In using f concerning body B, it is assumed that A exerts on B the ‘same force’ as it has exerted in those experiments. Let us consider why it is said, that ‘A exerts force’’ in all cases.

87 ‘‘Die Bewegungen und die Beschleunigungen sind Thatsachen, welche beobachtet werden ko¨nnen […] Wenn man dagegen von Kra¨ften spricht als den Ursachen dieser Bewegungserscheinungen, so weiß man von deren Wesen nichts weiter, als was man eben aus der Beobachtung des Bewegungsvorganges herauslesen kann […] Man kann daher von der Kraft nichts aussagen, was man nicht bereits von der Beschleunigung weiss’’ (p. 24).

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As Bergmann and Schaefer highlighted, the only sign we have of force is acceleration.88 Between both acceleration and force there is, however, a necessary connection, which is established by the law of inertia. The law states that a free body stays at rest or moves rectilinearly and uniformly. Hence, ‘free body’ is a sufficient condition for constant velocity. It follows from this implication that the moving body is not free if we observe non-constant velocity. This reasoning (‘free body’ implies ‘constant velocity’ and ‘nonconstant velocity’ therefore ‘non-free body’) is logically correct. It is the modus tollens: [(p ? q)^-q] ? -p. Thus, if the law of inertia is admitted, an accelerated motion requires an external something, which causes its acceleration. Coherently, we say that body A exerts force in all cases. In order to move on to ‘on body B’, we assume the ‘same force’. Let us try to express this dealing with the phenomena without the theoretical constraint derived from the law of inertia. Thanks to a set of experiments, f has been ascribed to body A. Then we assume, the body will move in the same way as it moved in those experiments. This assumption must be made if we want to use the information drawn from those experiments. Introducing f into the equation F = ma, where ‘m’ and ‘a’ refer now to B, we can predict some results. Thus, force f can be easily understood in conformity with our dealing with the phenomena. Let us turn now to the question asked in the introduction of how to conceive force in compliance with the contributions of Newton, Euler, Lagrange, etc. and without the inconveniences raised by the criticism of the concept. Thanks to Newton’s, Euler’s, Lagrange’s or Hertz’s work we learn that force can be conceived as a deviation from a certain motion. Taking each of these as a motion of reference and therefore force as a deviation from the motion of reference, the criticism of concept can be overcome. It is not necessary anymore to consider force as the cause of acceleration and to try to observe it. Our dealing with phenomena is clarified by the meaning of force as a value, which is ascribed to a body as a consequence of a certain set of experiments. All this enables us to understand the problems with the concept. D’Alembert’s and Carnot’s difficulties concern the lack of observability of force. What they could observe were motions. Nevertheless, they admitted real forces. The admission of what is not observable can be understood thanks to the law of inertia. Both accepted this law as the first statement of their theories. Thus, if they had not admitted the existence of force, their theories would not have been logically consistent. Tait, 1895, expressed the relationship between the law of inertia and the definition of force in the following clear way: ‘‘Thus, for the present, we have the definition of ‘‘force’’ as part of this First Law: -Force is whatever changes the state of the rest or uniform motion of a body’’ (p. 5). Even though the meaning of the law of inertia has changed, the structure of the law has been maintained. Hence, its logical consequence is still the same: acceleration requires force. As accelerated motions are observable, force must be there. The effects of force being observable, it must be a real existing thing. Thus, the spreading of the definition ‘force is the cause of acceleration’ is understandable, since the law of inertia has been accepted by almost all the authors (Coelho 2007).

88 ‘‘Das Tra¨gheitsgesetz sagt aus, dass ein Ko¨rper weder eine positive noch eine negative Beschleunigung erfa¨hrt, wenn keine a¨ußere Krafteinwirkung vorhanden ist. Beschleunigung ist also immer ein Anzeichen fu¨r das Vorhandensein einer solchen a¨ußeren Einwirkung, und zwar das einzige, das die Mechanik kennt’’ (p. 114).

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Saint-Venant’s and Mach’s theories of mechanics start from acceleration. In a second step, they define mass and finally force as a mere mathematical concept. That starting point of their theories is, however, incompatible with the classical system, for acceleration implies force, since the law of inertia is admitted. Hence, that sequence—acceleration, mass, force—could be and was criticized for presupposing force before introducing it (Roche 2006, p. 1030). This difficulty can now be overcome. If force is not anymore the real cause of acceleration, a definition of mass based on its measurement, which involves acceleration, is free of those objections (see Arons 1990, p. 51; Hecht 2006). Kirchhoff (1897) defined mechanics as the science of motion and planned to carry out a theory based on motion. He himself pointed out a difficulty: if there is a system of forces, it is not possible to determine the components only through motion. If force is understood as information drawn from some experiments, it follows that those components result from previous experiments. As these experiments are motions, Kirchhoff’s plan is free of that difficulty. Hertz’s solution for force shows the difficulty in connecting force with phenomena. This is corroborated by those authors like Hamel, Platrier or Ludwig, whose theses can be summed up by ‘force is not in real text’. This kind of thesis is now clear: as we can observe only acceleration, force cannot be seen in phenomena. As the determination of force is based on observations of the mechanical kind, our knowledge of phenomena is limited by the means employed in achieving it. Thus, it is understandable that Platrier 1954, said that the ‘‘profound cause’’ of the motions is unknown to us. Wilczek 2004, says ‘‘By comparison to modern foundational physics, the culture of force is vaguely defined, limited in scope, and approximate’’ (p. 12). In the next year, Wilczek characterizes the assumptions concerning force as ‘‘a sort of folklore’’ (2005, p. 10). The concept of force as a deviation from a certain motion enables us to integrate theses of philosophers of science and to understand their criticism. As the real cause of acceleration is a theoretical consequence of the law of inertia, it is understandable what Russell writes in Principles of Mathematics: ‘‘force is a mathematical fiction, not a physical entity’’ (1937, p. 482). Another mathematician and philosopher, Clifford wrote: ‘‘We do not know why the presence of one body tends to change the velocity of another; to say that it arises from the force resident in the first body acting upon the matter of the moving body is only to slur over our ignorance’’ (1955, p. 243). Nagel 1961, questions: ‘‘Why should uniform velocity be selected as the state of a body which needs no explanation in terms of the operation of forces, rather than uniform rest or uniform acceleration (such as motion along a circular orbit with constant velocity) […]?’’ (p. 177). As force has been taken as a deviation from a motion of reference and the uniform velocity is a characteristic of the motion of reference of the classical theory, there is no place for force there. As from the standpoint of the classical theory, there is force if the motion is not rectilinear, the circular motion implies force. We have, however, also seen that the circular uniform motion can be taken as a motion of reference. In fact, this had already been done by Lagrange’s formalism even though not verbalized. Hertz’s fundamental law also subsumes the circular motion. In sum, the ‘‘selection’’ referred to by Nagel depends on the motion of reference. As this does not have to be the motion of the law of inertia, the circular uniform motion can also be a motion of reference. Thus, the criticism is overcome and the criticized aspects are integrated. Ellis dealt with the concept of force in various papers (1962, 1963, 1965, 1976). His thesis, the law of inertia lays down the mechanical behavior of bodies and force is a matter of convention, was subject of intensive controversy (Hunt and Suchting 1969, pp. 235 ff).

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Let us consider this regarding the 1976 article: ‘‘it is a matter of convention what states we should regard as natural, and hence what things we should regard as force-effects, and hence what forces we should say exist’’ (p. 175). The concept of force as a deviation from a certain motion enables us an easy understanding of the thesis on the conventionality, since a motion of reference implies some choice. As it is not necessary for the motion of the law of inertia to be considered the natural motion, and neither for the other motions with an analogous function, it is not necessary to postulate the existence of forces anymore (Snider 1967). In discussing the topic ‘causal laws’, Chalmers 2008, writes: ‘‘Newton’s laws can readily be interpreted as causal laws describing the disposition of objects to exert and respond to specified forces. However, this is not the only way […] The laws of mechanics can also be written in a form that takes energy, rather than forces, as the starting point’’ (p. 223)’’. If force as the ‘cause of acceleration’ is a theoretical consequence of the law of inertia and ‘force exerted’ by an object on another expresses a piece of information drawn from other experiments, the Newtonian description of motion is not more ‘‘profound’’ than the Lagrangian one.

5 Some Educational Implications Students’ preconceptions or misconceptions and common sense beliefs concerning force have been the subject of much research (McClelland 1985; Halloun and Hestenes 1985; Bliss and Ogborn 1994; Hijs and Bosch 1995; Rowlands et al. 1999; de Lozano and Cardenas 2002 among many others). Teaching strategies and methods have also been developed (Arons 1990; Hestenes 1992; Rowlands et al. 1998; Stinner 2001; Galili 2001; Seker and Welsh 2006). All this deserves special attention.89 In what follows, only a few questions gathered from the literature will be dealt with. The relationship between force and motion has been the subject of many investigations and studies (Peters 1985; Halloun and Hestenes 1985; Galili and Bar 1992; Lombardi 1999; Carson and Rowlands 2005; Smith and Wittmann 2008). According to Rowlands et al. 2007: ‘‘‘misconceptions’ of force and motion are fundamental because understanding the Newtonian concept of force and motion is essential in understanding the system as a whole’’ (p. 31). One typical issue of this problem concerns the relationship between force and velocity, which is sometimes expressed as F = mv. This difficulty in learning could be overcome thanks to the concept of force as deviation from a certain motion. In this case, to speak of force already implies a motion. Without a motion of reference, force does not have any meaning. The idea of ‘‘deviation’’ from the motion of the law of inertia is used by Hestenes’ New Foundations for Classical Mechanics (1987, p. 589) (see also Arons 1990, p. 52). Carson and Rowlands 2005, write: ‘‘The problem is that we do not observe or experience ‘force’ as such’’ (p. 474). It is hence understandable that ‘‘it is difficult to see how force can be abstracted from experience’’ (p. 479). The forceless situation could be helpful in enabling us to make a comparison with situations with force. This situation is however unreal (Carson and Rowlands 2005, p. 483, 486–7). Let us consider if the difficulty could be overcome. 89 These issues deserve special attention because of the considerable amount of research literature. Some empirical educational research is also in preparation within the framework of the European Project History and Philosophy in Science Teaching.

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Let us take the linear oscillation of a spring as an example. It can be verified through observation that a body and a spring are involved in this motion. The motion of the body is accelerated as well as each element of the spring. If it is said that the spring exerts force f on the body and force is the cause of acceleration, we are led to looking for the cause of the acceleration there. However, it is difficult, perhaps impossible, to distinguish between cause and effect there, since both bodies are involved in the motion. If it is said that f was drawn from other experiments, it is not necessary to ‘‘see’’ force there. In general, if it is taught that force is there, where the motion is accelerated, a student will try to find in motion and through the observation of it, what does not come from there. If it is taught that force were gained from other experiments, the student will understand it without difficulty. In a study on scientific argumentation in the classroom, Driver et al. 2000, write: ‘‘As Kress, Ogborn, Jewitt, and Tsatsarleis (1998) pointed out nature does not ‘‘speak for itself,’’ particularly when the teacher is trying to convince pupils […] that objects continue in motion forever’’ (p. 291). Hanson 1965, highlighted the logical component of this problem. A frame of reference requires four particles. It is admitted in physics that any two particles attract each other. Thus it is impossible to determine how a ‘‘free body’’ moves (p. 14). Matthews 2008, put forward a radical question: ‘‘we never see force-free behaviour in nature, nor can it be experimentally induced, so what is the source and justification of our knowledge of bodies without impressed forces?’’ (p. 10) According to textbooks on mechanics, the law of inertia comes from Newton. It is perhaps ‘‘difficult’’ to accept a change of the status of a motion which has been adopted since the seventeenth century. However, a change in the law has already taken place, as we will see. According to Newton (1726), d’Alembert (1758), Laplace (1799), Carnot (1803), Poisson (1833) and many others, a ball on a flat table justifies the law of inertia. In fact, it can be observed that it stays at rest or moves rectilinearly and uniformly if it is not disturbed by an impressed force. The difficulty at that time was the uniformity, which could not be observed. For this reason, the staying at rest and moving rectilinearly were laws of nature in d’Alembert’s theory and the uniformity of motion was a corollary, as it could not be observed but only inferred. In contemporary mechanics, a ball on a flat table cannot however be used for the same aim. It is not a free body and the law is formulated for such a one. Nowadays, ‘‘free body’’ means a body without any constraint, whereas in the past, the body was free only in some directions and not in all thinkable ones. The meaning of ‘‘free body’’ differs, therefore, from ‘‘body not disturbed by impressed force’’. For this reason, it is now impossible to outline an experiment in compliance with the law, whereas the law was proved by experiments in the past. If we adopt contemporary statements of the law, we will have some problems, as has been pointed out. Considering the motion of the law of inertia as a motion of reference, the law of the past is integrated in its experimental component. Since the contemporary meaning of ‘‘free body’’ is avoided, it is not necessary to prove what cannot be. Furthermore, other statements with analogous function, which appeared in the course of the development of mechanics, such as Hertz’s fundamental law, can be integrated as well. In so far as the ascribing of a natural motion to bodies has led to the concept of force as a real something, the introducing of the motion of reference avoids the traditional problem with force.

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Author Biography Ricardo Lopes Coelho has been a ‘‘Professor Auxiliar’’ at the Faculty of Sciences of the University of Lisbon, since 1997, and a ‘‘Privatdozent’’ at the Technical University of Berlin, since 2001. He studied piano, philosophy and physics in Portugal, did his PhD at the TU-Berlin and his Habilitation in History and Philosophy of Exact Sciences at the same University. Among others, he published some articles concerning his main research interest, the understanding of scientific concepts and principles through its past and philosophy. He is the author of two books: Zur Konzeption der Kraft der Mechanik (On the Concept of Force in Mechanics) (2001) and O Conceito de Energia: Passado e Sentido (On the Concept of Energy: History and Meaning) (2006).

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